Spatial inversion in general relativity

I am unable to find any source describing at any lengths transformations of the type

[itex]x \rightarrow -x[/itex]

beyond the case of static spacetime (and even in the case of static spacetime, it is rarely the fundamental definition, which tends to be more along the lign of orthogonality with an hyperplane, [itex]\xi_{[a} \nabla_b \xi_{c]}[/itex]).

So my question is, if I have a space reversal symmetry, do the cross terms of that coordinates always vanish (if that coordinate also has a Killing vector at least), and is the most obvious proof ([itex]x \rightarrow -x, \ g_{xy} dx dy = -g_{xy} dx dy \rightarrow g_{xy} = 0[/itex] the correct one. Also, does it mean that all rotational symmetries have no cross terms, since they have [itex]\mathbb{Z}_2[/itex] as a subgroup?

If there is such a symmetry without a corresponding translation symmetry, does it just imply that the cross terms are odd functions?

You certainly have this symmetry locally in GR. Globally, there is not even any way to define the operation for an arbitrary spacetime. For example, there is not guaranteed to be a coordinate chart that covers all of spacetime, nor do you have any guarantee that the spacetime is orientable.